(Don't worry, it is not about the imperfect of this verb that I will ask about ). Here is my doubt: in tmesis or in composed verbs, ἀνα adds the notion of either "up" or "again". Right? I am not sure, then, how ἀνα modifies the sense of εἴρομαι. "Ask again"? But "ask again" wouldn't have sense here, because Odysseus is addressing Athena in the form of a young maiden whom he is supposed to see for the first time.

Geoffrey Steadman's notes say that τοῖον is an adverb here, but I always understand this word as a comparative one that requires an antecedent, and here I don't see the antecedent. Is somekind of anacoluthon? "Go in such silence...".

I wandered why is this infinitive in the perfect tense, and I found that Smyth says this about indirect discourse:

"Smyth #2019 wrote: Each tense of direct discourse is retained (with its proper meaning as regards stage of action) when it becomes infinitive in indirect discourse; but an imperfect is represented by the present infinitive; a pluperfect, by the perfect infinitive.

1. ἀνείρετο --Some compound verbs can be analyzed by breaking them down between the preverb and the basic verb, but in many cases, the preverb and verb can't be broken down this way. Homeric ἀνείρομαι just means "ask."

Hom. has many reduplicated second aorists, as πέ-πιθον from πείθω (πιθ-) persuade, κεκλόμην, κε-κλόμενος from κέλομαι command, λε-λαθέσθαι from λανθάνω (λαθ-) escape the notice of, πε-φιδέσθαι from φείδομαι (φιδ-) spare, ἤρ-αρον from ἀραρίσκω (ἀρ-) join, ὤρ-ορον from ὄρνυ_μι arouse. The indicative forms may take the syllabic augment, as in ἐ-πέ-φραδον from φράζω (φραδ-) tell. From ἐνίπτω chide and ἐρύ_κω check come ἠνί_παπον and ἐνένι_πον, and ἠρύ_κακον.